335 research outputs found
Lynden-Bell and Tsallis distributions for the HMF model
Systems with long-range interactions can reach a Quasi Stationary State (QSS)
as a result of a violent collisionless relaxation. If the system mixes well
(ergodicity), the QSS can be predicted by the statistical theory of Lynden-Bell
(1967) based on the Vlasov equation. When the initial distribution takes only
two values, the Lynden-Bell distribution is similar to the Fermi-Dirac
statistics. Such distributions have recently been observed in direct numerical
simulations of the HMF model (Antoniazzi et al. 2006). In this paper, we
determine the caloric curve corresponding to the Lynden-Bell statistics in
relation with the HMF model and analyze the dynamical and thermodynamical
stability of spatially homogeneous solutions by using two general criteria
previously introduced in the literature. We express the critical energy and the
critical temperature as a function of a degeneracy parameter fixed by the
initial condition. Below these critical values, the homogeneous Lynden-Bell
distribution is not a maximum entropy state but an unstable saddle point. We
apply these results to the situation considered by Antoniazzi et al. For a
given energy, we find a critical initial magnetization above which the
homogeneous Lynden-Bell distribution ceases to be a maximum entropy state,
contrary to the claim of these authors. For an energy U=0.69, this transition
occurs above an initial magnetization M_{x}=0.897. In that case, the system
should reach an inhomogeneous Lynden-Bell distribution (most mixed) or an
incompletely mixed state (possibly fitted by a Tsallis distribution). Thus, our
theoretical study proves that the dynamics is different for small and large
initial magnetizations, in agreement with numerical results of Pluchino et al.
(2004). This new dynamical phase transition may reconcile the two communities
Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions
We derive the exact expression of the diffusion coefficient of a
self-gravitating Brownian gas in two dimensions. Our formula generalizes the
usual Einstein relation for a free Brownian motion to the context of
two-dimensional gravity. We show the existence of a critical temperature T_{c}
at which the diffusion coefficient vanishes. For T<T_{c} the diffusion
coefficient is negative and the gas undergoes gravitational collapse. This
leads to the formation of a Dirac peak concentrating the whole mass in a finite
time. We also stress that the critical temperature T_{c} is different from the
collapse temperature T_{*} at which the partition function diverges. These
quantities differ by a factor 1-1/N where N is the number of particles in the
system. We provide clear evidence of this difference by explicitly solving the
case N=2. We also mention the analogy with the chemotactic aggregation of
bacteria in biology, the formation of ``atoms'' in a two-dimensional (2D)
plasma and the formation of dipoles or supervortices in 2D point vortex
dynamics
Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model
We present first elements of kinetic theory appropriate to the inhomogeneous
phase of the HMF model. In particular, we investigate the case of strongly
inhomogeneous distributions for and exhibit curious behaviour of the
force auto-correlation function and friction coefficient. The temporal
correlation function of the force has an oscillatory behaviour which averages
to zero over a period. By contrast, the effects of friction accumulate with
time and the friction coefficient does not satisfy the Einstein relation. On
the contrary, it presents the peculiarity to increase linearly with time.
Motivated by this result, we provide analytical solutions of a simplified
kinetic equation with a time dependent friction. Analogies with
self-gravitating systems and other systems with long-range interactions are
also mentioned
Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction
We study the relaxation of a test particle immersed in a bath of field
particles interacting via weak long-range forces. To order 1/N in the limit, the velocity distribution of the test particle satisfies a
Fokker-Planck equation whose form is related to the Landau and Lenard-Balescu
equations in plasma physics. We provide explict expressions for the diffusion
coefficient and friction force in the case where the velocity distribution of
the field particles is isotropic. We consider (i) various dimensions of space
and 1 (ii) a discret spectrum of masses among the particles (iii)
different distributions of the bath including the Maxwell distribution of
statistical equilibrium (thermal bath) and the step function (water bag).
Specific applications are given for self-gravitating systems in three
dimensions, Coulombian systems in two dimensions and for the HMF model in one
dimension
Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
We develop the kinetic theory of point vortices in two-dimensional
hydrodynamics and illustrate the main results of the theory with numerical
simulations. We first consider the evolution of the system "as a whole" and
show that the evolution of the vorticity profile is due to resonances between
different orbits of the point vortices. The evolution stops when the profile of
angular velocity becomes monotonic even if the system has not reached the
statistical equilibrium state (Boltzmann distribution). In that case, the
system remains blocked in a sort of metastable state with a non standard
distribution. We also study the relaxation of a test vortex in a steady bath of
field vortices. The relaxation of the test vortex is described by a
Fokker-Planck equation involving a diffusion term and a drift term. The
diffusion coefficient, which is proportional to the density of field vortices
and inversely proportional to the shear, usually decreases rapidly with the
distance. The drift is proportional to the gradient of the density profile of
the field vortices and is connected to the diffusion coefficient by a
generalized Einstein relation. We study the evolution of the tail of the
distribution function of the test vortex and show that it has a front
structure. We also study how the temporal auto-correlation function of the
position of the test vortex decreases with time and find that it usually
exhibits an algebraic behavior with an exponent that we compute analytically.
We mention analogies with other systems with long-range interactions
Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes
We complete our previous investigation concerning the structure and the
stability of "isothermal" spheres in general relativity. This concerns objects
that are described by a linear equation of state so that the
pressure is proportional to the energy density. In the Newtonian limit , this returns the classical isothermal equation of state. We consider
specifically a self-gravitating radiation (q=1/3), the core of neutron stars
(q=1/3) and a gas of baryons interacting through a vector meson field (q=1). We
study how the thermodynamical parameters scale with the size of the object and
find unusual behaviours due to the non-extensivity of the system. We compare
these scaling laws with the area scaling of the black hole entropy. We also
determine the domain of validity of these scaling laws by calculating the
critical radius above which relativistic stars described by a linear equation
of state become dynamically unstable. For photon stars, we show that the
criteria of dynamical and thermodynamical stability coincide. Considering
finite spheres, we find that the mass and entropy as a function of the central
density present damped oscillations. We give the critical value of the central
density, corresponding to the first mass peak, above which the series of
equilibria becomes unstable. Finally, we extend our results to d-dimensional
spheres. We show that the oscillations of mass versus central density disappear
above a critical dimension d_{crit}(q). For Newtonian isothermal stars (q=0) we
recover the critical dimension d_{crit}=10. For the stiffest stars (q=1) we
find d_{crit}=9 and for a self-gravitating radiation (q=1/d) we find
d_{crit}=9.96404372... very close to 10. Finally, we give analytical solutions
of relativistic isothermal spheres in 2D gravity.Comment: A minor mistake in calculation has been corrected in the second
version (v2
Gravitational instability of isothermal and polytropic spheres
We complete previous investigations on the thermodynamics of self-gravitating
systems by studying the grand canonical, grand microcanonical and isobaric
ensembles. We also discuss the stability of polytropic spheres in the light of
a generalized thermodynamics proposed by Tsallis. We determine in each case the
onset of gravitational instability by analytical methods and graphical
constructions in the Milne plane. We also discuss the relation between
dynamical and thermodynamical stability of stellar systems and gaseous spheres.
Our study provides an aesthetic and simple approach to this otherwise
complicated subject.Comment: Submitted to A&
Gravitational instability of slowly rotating isothermal spheres
We discuss the statistical mechanics of rotating self-gravitating systems by
allowing properly for the conservation of angular momentum. We study
analytically the case of slowly rotating isothermal spheres by expanding the
solutions of the Boltzmann-Poisson equation in a series of Legendre
polynomials, adapting the procedure introduced by Chandrasekhar (1933) for
distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood
(1967) in the temperature-energy plane is deformed by rotation. We find that
gravitational instability occurs sooner in the microcanonical ensemble and
later in the canonical ensemble. According to standard turning point arguments,
the onset of the collapse coincides with the minimum energy or minimum
temperature state in the series of equilibria. Interestingly, it happens to be
close to the point of maximum flattening. We determine analytically the
generalization of the singular isothermal solution to the case of a slowly
rotating configuration. We also consider slowly rotating configurations of the
self-gravitating Fermi gas at non zero temperature.Comment: Submitted to A&
- …